Sum of Consecutive Terms of Pell and Related Sequences

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by Pₙ := 2Pₙ₋₁+Pₙ₋₂ for n≥ 2 and P₀=0, P₁=1, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if 4 N. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for n≥ k+1, with p(0)=p(1)=⋯ =p(i)=0 and p(i+1)=⋯ = p(k)=1 for 0≤ i≤ k-1, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when N and k are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences.